Theorem cusgrres 29420

Description: Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)

Hypotheses

Ref	Expression
cusgrres.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrres.e	⊢ 𝐸 = (Edg‘𝐺)
cusgrres.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
cusgrres.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
cusgrres	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem cusgrres

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrusgr 29390	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
2		cusgrres.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		cusgrres.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
4		cusgrres.f	. . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
5		cusgrres.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
6	2, 3, 4, 5	usgrres1 29286	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)
7	1, 6	sylan 580	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph)
8		iscusgr 29389	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
9		usgrupgr 29156	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ UPGraph)
11	10	anim1i 615	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉))
12	11	anim1i 615	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})))
13	2	iscplgr 29386	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺)))
14		eldifi 4079	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ∈ 𝑉)
15	14	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → 𝑣 ∈ 𝑉)
16		eleq1w 2812	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (UnivVtx‘𝐺) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
17	16	rspcv 3571	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
18	15, 17	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁}))) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺)))
19	18	ex 412	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝐺))))
20	19	com23 86	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (∀𝑛 ∈ 𝑉 𝑛 ∈ (UnivVtx‘𝐺) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
21	13, 20	sylbid 240	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))))
22	21	imp 406	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) → ((𝑁 ∈ 𝑉 ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺)))
23	22	impl 455	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝐺))
24	2, 3, 4, 5	uvtxupgrres 29379	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → (𝑣 ∈ (UnivVtx‘𝐺) → 𝑣 ∈ (UnivVtx‘𝑆)))
25	12, 23, 24	sylc 65	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → 𝑣 ∈ (UnivVtx‘𝑆))
26	25	ralrimiva 3122	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
27	8, 26	sylanb 581	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆))
28		opex 5402	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
29	5, 28	eqeltri 2825	. . . 4 ⊢ 𝑆 ∈ V
30	2, 3, 4, 5	upgrres1lem2 29282	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
31	30	eqcomi 2739	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
32	31	iscplgr 29386	. . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
33	29, 32	mp1i 13	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 ∈ ComplGraph ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ (UnivVtx‘𝑆)))
34	27, 33	mpbird 257	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplGraph)
35		iscusgr 29389	. 2 ⊢ (𝑆 ∈ ComplUSGraph ↔ (𝑆 ∈ USGraph ∧ 𝑆 ∈ ComplGraph))
36	7, 34, 35	sylanbrc 583	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ ComplUSGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ∉ wnel 3030  ∀wral 3045  {crab 3393  Vcvv 3434   ∖ cdif 3897  {csn 4574  ⟨cop 4580   I cid 5508   ↾ cres 5616  ‘cfv 6477  Vtxcvtx 28967  Edgcedg 29018  UPGraphcupgr 29051  USGraphcusgr 29120  UnivVtxcuvtx 29356  ComplGraphccplgr 29380  ComplUSGraphccusgr 29381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-dju 9786  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-n0 12374  df-xnn0 12447  df-z 12461  df-uz 12725  df-fz 13400  df-hash 14230  df-vtx 28969  df-iedg 28970  df-edg 29019  df-uhgr 29029  df-upgr 29053  df-umgr 29054  df-uspgr 29121  df-usgr 29122  df-nbgr 29304  df-uvtx 29357  df-cplgr 29382  df-cusgr 29383

This theorem is referenced by:  cusgrsize  29426